Exercise 1, § 3

Define an increasing function $g$, admitting a continuous derivative on a neighbourhood of $+\infty$, such that $g$ and $1/x$ are comparable to order 1, but that $x$ and $1/g$ are not comparable to order 1 (take $g'(x) = 1$ except on sufficiently small intervals with centres at the points $x = n$ ($n$ an integer $> 0$) on which $g'$ takes very large values).